Okay. In this video
I want to talk about a little bit of
geometry and two special right triangles. They're going to be the 45 45 90
degree right triangle,
where one angle is 90 and the other two angles are each 45 degrees and then the other triangle
is going to be one with a right angle of 90 degrees,
nd then let's suppose one angle is 30 degrees and another one is 60 degrees.. So forgive my poor artistry.
I definitely didn't use a protractor or anything
more accurate than my bad
picture, so
hopefully everything here is clear.
Let's look at the 45-45-90
triangle first, and let's suppose that opposite one of the angles
that has that has degree measure of 45,
let's suppose that the length of the side of the triangle is a.
Well, if you think about it, if [the side] opposite
the angle of 45 degrees is a, that means this other side is also going to have to have length a
because it's also opposite an angle with degree measure 45 degrees. We would like to figure out this missing side.
I don't know maybe we'll call it
generically c. We could simply use the Pythagorean theorem here, and remember the Pythagorean theorem says take one side and square it,
take the other side and square it, and we set that equal to the hypotenuse squared.
We've got one a squared plus one a squared that's simply 2 a squared and
What we're trying to do here is simply solve for c.
And to solve for c, we can take the square root of both sides.
Normally when you take square roots you get positives and negatives, but clearly c has to be a positive number.
Remember, we can break this up as the square root of two
times the square root of a squared, and the square root of a squared is just a.
So it says that, if we know that that this side is length a, and this side is length a, it
says this other side would have] to be the square root of 2 a.
That just gives us a nice little relationship about the values...
about the values on this triangle. So for example, let me do one here real quick. Let's make up a couple.
Suppose we knew that this had length 3. Suppose this was a 45 degree,
45 degree, and then we have 90 degrees. All this result says [is] suppose
this side is 3, well, by default that would make this side 3. It says to get the the
hypotenuse, you simply take that length, and it says you multiply it by the square root of 2, and, lo and behold,
hey, we've got the length of the hypotenuse.
Likewise,
suppose this had been 10. If this had length 10,
This side would also have length 10, and it says this hypotenuse, then, would simply have value...
would have a length of 10 times the square root of 2. So a nice ... a nice little simple result here for sure.
Let's maybe do one backwards.
Suppose we know the length of the hypotenuse. Suppose it's a nice number ...
How about, I don't know, how about 8. That's a nice whole number.
Well, we want to figure out the missing side.
Okay, so let's maybe call the missing side a
Well, we know that
this ... this opposite side is going to have length
square root of 2 times a. That would be the length of the hypotenuse.
But we also know in this case that that has to equal 8.
So if we divide both sides... Again, we're trying to solve for a. If we divide both sides by the square root of 2,
we would simply get that a
equals 8 over
square root of 2. And a lot of people don't like to have radicals in the denominator,
so certainly it would be correct to say that this has length
8 over square root of 2. A lot of people don't like that. So what we'll do is we'll rationalize the denominator. You could multiply
the top and the bottom of our fraction by square root of 2.
We're not going to do anything in the numerator; that'll just give us 8
times square root of 2.
On the bottom, though, remember square root of 2 times square root of 2 is going to be the square root of 4.
But the square root of 4 is simply 2, so I usually actually skip that step.
To me,
I always thought a square root times the square root just gets rid of the square root - that's what it does. And now since this is all
multiplication, we can divide. We have 8 divided by 2,
which would be 4
square root of 2. So,
equivalently, instead of 8 over square root of 2, we can simply say that this has length 4 square root of 2, and
this also has length 4 square root of 2.
And we
Figured out the sides ... the links of the missing sides. So no problem here.
Let's ...
Let's look at our
30-60-90 triangle here real quick.
And what I'm going to do to justify this one is I'm actually going to make it a little bit bigger.
So this is 30 degrees.
Let's make this 60 degrees. Let me make this a little bit bigger...
Let's suppose this angle inside is 30 degrees, and let's simply suppose that this bottom has length, again, a like we had
over here.
What I'm going to do is I'm going to actually stick another copy of it...
I'm going to imagine reflecting this and just kind of sticking it on the other side. so kind of a mirror image of that triangle.
Well, if this is length a, since we're just going to make a little mirror image of it, this would also be length a, and,
since this was 60 degrees, this would also be 60 degrees.
Again, these are both little right triangles. If you think about the length of this big triangle now,
well, this is length a, this is length a.
So this whole bottom length is 2a.
But let's make one more observation here as well.
If this is 30 degrees, that means this part up here is also 30 degrees, and,
well, that really now tells us that this whole angle in the big triangle...
That whole angle in the big triangle would have to be 60 degrees because it's 30 plus 30.
Well, what our triangle ... what we basically have now, it says, if you have a side, that's opposite
an angle with measure of 60 degrees, it says that has length 2a.
Well, if that has length
2a, ... (again, let's look at our triangle here) ...Well, this is sixty degrees.
That means the side opposite that ...opposite to that
would also have length 2a, and this would also be 2a.
Okay, so what we did was we simply took this triangle and made a little mirror reflection of it,
and we just made some observations here.
So now we know that if this is ... if the length opposite 30 degrees is a,
we know the hypotenuse by default has length 2a, and again now
we're just going to use Pythagorean Theorem to figure out this missing... this missing height.
So by Pythagorean Theorem, it says we would get a squared plus ... maybe I'll call the missing side b...
It says a squared plus b squared...
that would equal the hypotenuse, which is 2a, squared.
And, again, we're going to try to figure out this missing length b. So it says we get a squared plus b squared...
2a times 2a it's going to be 4a squared.
If we subtract a squared from both sides we'll get b squared equals
3 a squared. And the same thing as before; we'll take the square root of both sides.
We'll get that b is going to be the square root of 3 times a.
So this missing length over here ... if this is a, this is 2a, this is going to be
a times square root of 3. So it's going to be the ultimate ... the ultimate little result.
But I think it's always nice to kind of be able to justify things because then you're not memorizing;
you're understanding where things come from, and that way too, on a test or something like that, if you forget the formula,
you can ... you can figure it out.
So let me maybe just do one or two here real quick.
Again, just kind of a little introduction here to do these types of triangles. So let's suppose this is 30 degrees,
60 degrees,
90 degrees.
Suppose we know that this, the hypotenuse, has length ...
Let's even make it easier. Suppose the side opposite 30 degrees ...
Suppose that has length 5. (That's kind of the, I think, the most simple one we can do.)
And it says if you know the side
opposite 30 degrees, to get the hypotenuse,
we just double that. That's what this result says. So we'll take 2 times 5, which is 10.
And then it says ,to get the the length of the triangle opposite 60 degrees, we just take this number and
we simply multiply it by square root of 3. So easy enough...
Let's maybe
do two others real quick.
Same thing. Suppose (suppose) we knew the hypotenuse. Suppose it had value...
I don't know ... how about 30.
Well, it says, to get the side
opposite 30 degrees, we would just divide that by 2. So 30 divided by 2 is simply 15. And once we know the side
opposite 30 degrees, again, we just multiply that by
square root of 3 to get the missing side.
Let's maybe do one more real quick again. You know, nothing ...
nothing super exciting. I don't know maybe let's get your heart racing. I don't know that
it does mine, but certainly these are very useful little tricks ... little geometry tricks ...these ... these are
triangles that seem to crop up.
So I think useful little results. Let's suppose we know the side opposite 60 degrees. Suppose
we know that that is equal to ... let's just make up a number...
How about it's equal to 8?
Okay, the same thing. Well, we could call this this length a. Let's find this side first.
Well, if this is a, again, we know that this other side would have to equal a times square root of 3.
So to figure out our missing length, a, we can just divide both sides by square root of 3.
We'll get 8 over root 3. That's going to be a. Again, that would be perfectly acceptable
to me, but some people, again, don't like radicals in the denominator.
So if we multiply top and bottom by square root of 3
(there's really not much good simplification in this case), we would just get 8
square root of 3 on top, and then square root of 3 times square root of 3 is simply 3.
You know, 8 over 3 doesn't reduce nicely so I would probably leave it like that.
So I think we started off with this having length 8. That means this side would have length 8 square root of 3
over 3, and...
Yeah, so not, you know, not nice whole numbers. But this is kind of the point of why I picked these problems.
You're not always going to get nice numbers.
Lastly, if we know the side opposite 30 degrees, all we have to do is double that to get the hypotenuse.
So if we take 2 times 8 Square root of 3 over
3...
Again, we can think about 2 as simply being 2 over 1 to multiply fractions...
We just multiply across the top, so 2 times 8 would be 16
square root of 3, and then, on the bottom, 1 times 3 is simply going to give us 3. So we know all of our...
Again, now we know all of our missing values by knowing only one of the values.
So definitely kind of tedious little stuff... nothing too crazy though.
Again, I think if you can remember the derivation, it'll certainly, again, now your understanding, and if you do forget these formulas because I
forget them, for sure all the time. If you do forget them,
I think you'll be able to ... be able to reproduce them relatively quickly. So all right.
I hope this helps. As always, if there's any questions or comments, feel free to post them.